General Physics I Work & Energy

Forms of Energy Kinetic: Energy of motion. A car on the highway has kinetic energy. We have to remove this energy to stop it. The brakes of a car get HOT! This is an example of turning one form of energy (motion) into another (thermal energy).

Energy Conservation Energy cannot be destroyed or created. Just changed from one form to another. We say energy is conserved! True for any closed system. When we put on the brakes, the kinetic energy of the car is turned into heat using friction in the brakes. The total energy of the “car-brakes-road-atmosphere” system is the same. The energy of the car “alone” is not conserved... It is reduced by the braking. Doing “work” on an isolated system will change its “energy”!

Definition of Work: Units: Force x Distance = Work Ingredients: Force (F), displacement (x) Work, W, of a constant force F acting through a displacement x is: W = F x = F x cos  = Fx x F  Fx x Units: Force x Distance = Work displacement Newton x Meter = Joule

Non-Conservative Work If the work done does not depend on the path taken, the force is said to be conservative. If the work done does depend on the path taken, the force is said to be non-conservative. The most common example of a non-conservative force is friction. Thus, the work done by friction in pushing an object a distance D is given by: Wf = -f • D NOTE: The work done by friction is ALWAYS negative (opposite the direction of motion)

WTOT = FTOT x It’s the total force that matters!! Work on an Incline A large red box is pulled up a rough (m > 0) incline by a rope-pulley-weight arrangement as shown below. How many forces are doing work on the red box? (a) 2 (b) 3 (c) 4 WTOT = FTOT x It’s the total force that matters!! Is work done by N?

Work & Kinetic Energy A force F pushes a box across a frictionless floor for a distance x. The speed of the box is vo before the push and vf after the push. Since the force F is constant, acceleration a will be constant. We have shown that for constant a: vf2 - vo2 = 2a(xf -xo) = 2ax (one of our original equations of motion). Multiply by 1/2m: 1/2mvf2 - 1/2mvo2 = max But F = ma 1/2mvf2 - 1/2mvo2 = Fx = WF x F vo vf m a x

Work & Kinetic Energy... So we found that: 1/2mvf2 - 1/2mvo2 = F*x = WF Define Kinetic Energy, KE: KE = 1/2mv2 KEf - KEi = WF WF = KE (Work/kinetic energy theorem) {Net Work done on object} = {Change in kinetic energy of object}

Work & Energy Two blocks have masses m1 and m2, where m1 > m2. They are sliding on a frictionless floor and have the same kinetic energy when they encounter a long rough stretch (i.e. m > 0) which slows them down to a stop. Which one will go farther before stopping? HINT: Think about the relation that: KE = WF = Fx (a) m1 (b) m2 (c) They will go the same distance m1 m2

Energy Lost to Friction In situations where there is non-zero friction we’ve seen where, from a force perspective, that motion is slowed. From an energy perspective that “loss of motion” cannot simply be destroyed…it must go somewhere. In general: Eμ = μ N x For a level surface: Eμ = μ m g x For an incline: Eμ = μ m g cos(θ) x

A simple application: Work done by gravity on a falling object What is the speed of an object after falling a distance H, assuming it starts at rest? v0 = 0 H

Potential Energy For any conservative force we can define a potential energy function PEg such that: The potential energy function PEg is always defined only up to an additive constant (PEg = mgh) The potential energy is based on the position of an object rather than any motion (kinetic energy) that it may have. You can choose the location where PEg = 0 to be anywhere convenient.  PEg = PEgf – PEgi = -Wnet

More about the Conservation of Energy If only conservative forces are present, the total kinetic plus potential energy of a system is conserved, i.e. the total “mechanical energy” is conserved. (note: E = Emechanical throughout this discussion) E = KE + PE is constant!!! Both KE and PE can change, but E = KE + PE remains constant. But we’ve seen that if non-conservative forces act then energy can be dissipated into other modes (thermal, sound, etc.) E = KE + PE E = KE + PE = W + PE = W + (-W) = 0 using KE = W using PE = -W Power is the “rate of doing work”:

Falling Objects Three objects of mass m begin at height h with velocity 0. One falls straight down, one slides down a frictionless inclined plane, and one swings on the end of a pendulum. What is the relationship between their velocities when they have fallen to height 0? v=0 vi H vp vf Free Fall Frictionless incline Pendulum (a) Vf > Vi > Vp (b) Vf > Vp > Vi (c) Vf = Vp = Vi

Example: The simple pendulum Suppose we release a mass m from rest a distance h1 above its lowest possible point. What is the maximum speed of the mass and where does this happen? To what height h2 does it rise on the other side? m h1 h2 v

Problem: Hotwheel A toy car slides on the frictionless track shown below. It starts at rest, drops a distance d, moves horizontally at speed v1, rises a distance h, and ends up moving horizontally with speed v2. Find v1 and v2. v2 d h v1

Potential Energy & The Spring For a spring we know that Fx = -kx. F(x) xo xf x Change in the potential energy of a spring (PEs) relaxed position -kx F = - k xo F = - k xf

Conservation of Energy & The Spring A block slides on a horizontal frictionless surface with a speed v. It is brought to rest when it hits a bumper that compresses a spring. How much is the spring compressed? An object is released from rest at a height H on a curved frictionless ramp. At the foot of the ramp is a spring of constant k. The object slides down the ramp and into the spring, compressing it a distance x before coming momentarily to rest. What is x?

End of Work & Energy Lecture